Finding a few solutions for a given problem that are diverse, as opposed to finding a single best solution to solve the problem, has recently become a notable topic in theoretical computer science. Recently, Baste, Fellows, Jaffke, Masařík, Oliveira, Philip, and Rosamond showed that under a standard structural parameterization by treewidth, one can find a set of diverse solutions for many problems with only a very small additional cost [Artificial Intelligence 2022]. In this paper, we investigate a much stronger graph parameter, the cliquewidth, which can additionally describe some dense graph classes. Broadly speaking, it describes graphs that can be recursively constructed by a few operations defined on graphs whose vertices are divided into a bounded number of groups while each such group behaves uniformly with respect to any operation. We show that for any vertex problem, if we are given a dynamic program solving that problem on cliquewidth decomposition, we can modify it to produce a few solutions that are as diverse as possible with as little overhead as in the above-mentioned treewidth paper. As a consequence, we prove that a diverse version of any MSO$_1$ expressible problem can be solved in linear FPT time parameterized by the cliquewidth, the number of sought solutions, and the number of quantifiers in the formula, which was a natural missing piece in the complexity landscape of structural graph parameters and logic for the diverse problems. We prove our results allowing for a more general natural collection of diversity functions compared to only two mostly studied diversity functions previously. That might be of independent interest as a larger pool of different diversity functions can highlight various aspects of different solutions to a problem.

翻译：在给定问题中寻找若干多样化解（而非单一最优解）已成为理论计算机科学领域近期备受关注的研究方向。最近，Baste、Fellows、Jaffke、Masařík、Oliveira、Philip与Rosamond的研究表明，在树宽这一标准结构参数化框架下，仅需付出极小额外代价即可为众多问题找到多样化解集[《人工智能》2022]。本文研究一种更强的图参数——团宽，该参数还能描述某些稠密图类。概言之，团宽描述的图可通过有限操作递归构造，这些操作定义在顶点被划分为有限组别的图上，且每个组别在任何操作下均保持行为一致性。我们证明对于任意顶点问题，若已获得基于团宽分解的动态规划解法，则可通过改进算法以极小开销（与上述树宽研究中的开销相当）生成尽可能多样化的解集。由此推论，任何MSO$_1$可表达问题的多样化版本均可在线性FPT时间内求解，其参数包括团宽、目标解数量及公式中的量词数量，这填补了结构图参数与逻辑在多样化问题复杂度版图中长期缺失的关键环节。相较于先前主要研究的两种多样性函数，我们的证明允许使用更广泛的自然多样性函数集合，这或具有独立研究价值，因为更丰富的多样性函数库能凸显问题不同解决方案的多元特征。